1. On a Production/Inventory System with Strategic Customers and Unobservable Inventory Levels Can Oz Fikri Karaesmen SMMSO 2015 3 June 2015

2. Agenda Introduction Model definition Solution Procedure Computational Study Future Research

3. Introduction On a Production/Inventory System with Strategic Customers and Unobservable Inventory Levels

4. Introduction On a Production/Inventory System with Strategic Customers and Unobservable Inventory Levels Decision-making entities Risk neutral Fixed reward vs. Waiting cost

5. Introduction On a Production/Inventory System with Strategic Customers and Unobservable Inventory Levels Inventory position of the system is not shared with the customers, only production target and service rate

6. Introduction On a Production/Inventory System with Strategic Customers and Unobservable Inventory Levels Fixed service rate server Fixed payment from joining customers vs. Inventory holding cost

7. Literature Strategic customers in queueing systems Naor, P. 1969 Edelson, N.M., D.K. Hildebrand 1975 Hassin, R., M. Haviv 2003 Strategic customers in make-to-stock systems

8. Model Definition Strategic customers arrive at an inventory system that is operated by a base-stock policy Customer’s joining decision is based on other customers’ decisions Producer is the leader in the Stackelberg game and acts knowing customers’ decisions

9. Different versions Customer types Exogenous customers vs. strategic customers Homogenous vs. heterogeneous customers Demand Single unit vs. multiple unit Partial vs. full batches Information Observable vs. unobservable queue length Exit strategy Balking vs. staying Service quality Perfect vs. stochastic service quality

10. Different versions Customer types Exogenous customers vs. strategic customers Homogenous vs. heterogeneous customers Demand Single unit vs. multiple unit Partial vs. full batches Information Observable vs. unobservable queue length Exit strategy Balking vs. staying Service quality Perfect vs. stochastic service quality

11. Model Notation Customer’s problem Poisson arrivals(rate λ ) with unit demand Unobservable queue length, Reward for finished service, R-p Waiting cost per unit time, c May not join the system (customer’s decision) Will not leave the system after joining Producer’s problem Exponential production time with rate μ Revenue per customer, p Holding cost per unit time, h No backordering or waiting cost Sets the production limit S (producer’s decision) Customers and producer studied Buzacott&Shanthikumar 1993 and were very good students in stochastic models course

12. Order of Events Customers know R, c, p and λ. Producer announces target inventory level S and service rate μ. Customers decide on their individual joining probability q. Producer knew the joining probability and set S to maximize his profit.

13. M/M/1 queue Our System Equivalent System

14. Useful Results where E[W] is the expected waiting time and E[I] is the expected inventory level q is the joining probability S is the production limit Buzacott&Shanthikumar 1993

15. Considering expected waiting time and the reward, each customer makes a decision Customer’s Decision From queueing counterpart we know equilibrium joining probability is 0 when All customers might join the system ifOther than these cases equilibrium joining probability is unique and solves Equation (I)

16. Customer’s Decision Let’s assume q + >q is the joining probability, then some customers will left since their reward is negative Similarly if q - <q percent is the equilibrium joining probability then some customers will increase their joining probability

17. Customer’s Joining Rate

18. q S is a non-decreasing function of S After a threshold level joining probability is 1 Equilibrium Joining Probability

19. Tries to maximize his profit by setting the inventory target S where the expected inventory level is Producer’s decision set is bounded Producer’s Problem

20. Step 1:Set S = 0, calculate the equilibrium joining rate and resulting profit for the producer Step 2: If q S ≠ 0, set S = S+1 go to Step 1 else go to Step 3 Step 3: Find the maximizer of the expected profit among the calculated

21. Tries to maximize total system profit by deciding on arrival rate λ and inventory target S For fixed λ, optimal S is the solution of Equation (II) (Buzacott&Shanthikumar 1993) Social Optimization

22. p ~ DU[1,20] and λ ~ CU[0,0.97] Lower profitability R/c=2 vs higher profitability R/c=10 Customer’s willingness to join is increasing Computational Study SettingLow R/cHigh R/cR/c<p/hR/c=p/hR/c>p/h hpCU[0,1]pCU[0,1]/5 Rp+DU[1,20] p+DU[1,10]p+DU[1,20]p+DU[1,40] cRCU[0,1]RCU[0,1]/5

23. Joining rate comparison

24. Target inventory level comp.

25. Profit comparison

26. Profit comparison cont’d MetricLow R/cR/c<p/hR/c=p/hR/c>p/h T( λ D,S D )/ T( λ C,S C ) 0.720.910.850.79 Π ( λ D,S D )/ Π ( λ C,S C ) 1.100.970.90.83 θ ( λ D,S D )/ θ ( λ C,S C ) 1.231.141.31.54 % of instances customer loss391253

27. Conclusion Developed a make-to-stock production system with strategic customers Characterized customers’ equilibrium joining probability and producer’s optimal decision Identified cases where centralization is profitable

28. Future Research Fully observable system Finished the analysis of centralized and decentralized problems Showed the concavity of producer’s profit function Showed the optimality of base stock policy and Partially observable system Comparison with pure queueing system